8/10/2019 MIT complex RES_18_008_partI_lec02
1/11
Study Guide
Block 1: Rn
Introduction
to Functions of a omplex
Variable
U n i t :
Comulex
Functions of a Cm l e x Variable
1
overview
The
main
a i m
of this
Black is
to study t h e
calculus
of functions
of
a complex variable.
Jus t
as
was th case
when
we studied real
variables,
our approach is f i r s t to
discuss
the nunber system
and then to
apply
the
limit: concept to these functions.
In ur f i r s t
three u n i t s
we have
developed th
complex number
system
in some deta i l and you should
now
feel a
b i t
more
at
home with
the concept
of this number s y s t e m In this un i t as
our title
implies,
we
shall
di sc us s
functions
def ined
on the
complex numbers;
and
the remainder of
the Block will then be
devoted to the
ttrpics
usually
ide nt i f i e d w i t h
calculus.
You will nutice at
there is
no lecture for this
uni t .
The
reason is
t h a t
from a
gaametrical
point of view, as
we shal l
show
in
the exercises,
tha
study
of complex functions
of
a
complex variable is equivalent to the
seal
problem
of
describing
mappings of the =-plane into the uv-plane. The
only
di f ference ,
in t:ermsof the
language
of the Argand diagram, is tha t
the
xy
plane
becomes known as the
s-plane
while t h e uv-plane becomes
known
as
the w-plane.
Then
in
a
way
analogous
to
th
notation
of writing y
f x ) in the
study of
real
functions of a real
variable,
we write w
2 ) when we are studying a compler
f netiona of a
complex variable,
Skim
Thomas,
Section 19.3, A f t e r th exercises you may
wish
to
re-read t is
section
in
greater
d e t a i l ,
but our
i n i t i a l
aim
is f o r you to read just enough
to get
a quick overview of what
this unit deals w i t h Then you
should
proceed dicactly to the
exercises
since
our
fee l ing is
that
the best
way
to learn this
topic
is
in
t erm of working
speci f ic
exercises.
If
there
are
s t i l l certain points bothering you after you have
completed
this
unit,you way be heLped
by the
lecture of t e following
uni t which begins with a review of t h e concepts in this
u n i t
